# Matrix factorization method

*matrix forward-backward substitution method*

A method for solving finite-difference systems that approximate boundary value problems for systems of ordinary differential equations in one-dimensional problems, and for elliptic equations in two-dimensional problems.

The solution of the three-point difference scheme

where is an unknown grid vector, is the right-hand side vector and are given square matrices, under the boundary conditions

is sought for, as in the scalar case, in the form

(*) |

The coefficients (the matrix and the vector ) are determined from the recurrence relations ( "forward substitution" )

while and are given by the left boundary condition:

The are calculated by formula (*) ( "backward substitution" ), and

There is stability of this method to rounding errors under the conditions

which implies that , (see [1]). A different form of the stability conditions is also available (see [2], [3]). The matrix factorization method is applied also to two-point difference schemes (see [3]). A variant in which inversion of matrices is replaced by orthogonalization is also used (see [4]).

#### References

[1] | A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian) |

[2] | V.V. Ogneva, "The "sweep" method for the solution of difference equations" USSR Comp. Math. Math. Phys. , 7 : 4 (1967) pp. 113–126 Zh. Vychisl. Mat. i Mat. Fiz. , 7 : 4 (1967) pp. 803–812 |

[3] | A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian) |

[4] | S.K. Godunov, "A method of orthogonal successive substitution for the solution of systems of difference equations" USSR Comp. Math. Math. Phys. , 2 : 6 (1962) pp. 1151–1165 Zh. Vychisl. Mat. i Mat. Fiz. , 2 : 6 (1962) pp. 972–982 |

[5] | E.L. Wachspress, "Iterative solution of elliptic systems and applications to the neutron diffusion equations of reactor physics" , Prentice-Hall (1966) |

#### Comments

#### References

[a1] | I. [I. Babushka] Babuška, M. Práger, E. Vitásek, "Numerical processes in differential equations" , Interscience (1966) |

**How to Cite This Entry:**

Matrix factorization method.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Matrix_factorization_method&oldid=15766